Does the CIA have jurisdiction in the us?

Generally speaking, No, they get into hot water if they do- like the Watergate. When the agency was formed the idea was to have it headquartered in Chicago, as that is in Central TIME ZONE. This may or may not be true. FBI has many Us Field offices- equivalent to precincts with regular police. an Inspector in the FBI ( and KGB) is in charge of a Field Office and is the equivalent of Major in the Army or AF.